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1、August7,201221:05c08Sheetnumber1Pagenumber451cyanblackCHAPTER8NumericalMethodsUptothispointwehavediscussedmethodsforsolvingdifferentialequationsbyusinganalyticaltechniquessuchasintegrationorseriesexpansions.Usually,theemphasiswasonfindinganexactexpressionforthesol
2、ution.Unfortunately,therearemanyimportantproblemsinengineeringandscience,especiallynonlinearones,towhichthesemethodseitherdonotapplyorareverycomplicatedtouse.Inthischapterwediscussanalternativeapproach,theuseofnumericalapproximationmethodstoobtainanaccurateapprox
3、imationtothesolutionofaninitialvalueproblem.Wepresentthesemethodsinthesimplestpossiblecontext,namely,asinglescalarfirstorderequation.However,theycanreadilybeextendedtosystemsoffirstorderequations,andthisisoutlinedbrieflyinSection8.5.Theproceduresdescribedherecanbeex
4、ecutedeasilyonpersonalcomputers.8.1TheEulerorTangentLineMethodTodiscussthedevelopmentanduseofnumericalapproximationprocedures,wewillconcentratemainlyonthefirstorderinitialvalueproblemconsistingofthedifferentialequationdy=f(t,y)(1)dtandtheinitialconditiony(t0)=y0.(
5、2)Weassumethatthefunctionsfandfyarecontinuousonsomerectangleinthety-planecontainingthepoint(t0,y0).Then,byTheorem2.4.2,thereexistsauniquesolutiony=φ(t)ofthegivenprobleminsomeintervalaboutt0.IfEq.(1)isnonlinear,then451August7,201221:05c08Sheetnumber2Pagenumber452c
6、yanblack452Chapter8.NumericalMethodstheintervalofexistenceofthesolutionmaybedifficulttodetermineandmayhavenosimplerelationshiptothefunctionf.However,inallourdiscussionsweassumethatthereisauniquesolutionoftheinitialvalueproblem(1),(2)intheintervalofinterest.InSecti
7、on2.7wedescribedtheoldestandsimplestnumericalapproximationmethod,namely,theEulerortangentlinemethod.Toderivethismethod,letuswritethedifferentialequation(1)atthepointt=tnintheformdφ(tn)=f[tn,φ(tn)].(3)dtThenweapproximatethederivativeinEq.(3)bythecorresponding(forw
8、ard)differencequotient,obtainingφ(tn+1)−φ(tn)∼=f[t,φ(t)].(4)nntn+1−tnFinally,ifwereplaceφ(tn+1)andφ(tn)bytheirapproximatevaluesyn+1andyn,respectively,andsolvef
